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University of Cambridge

1. Chen, Cangxiong. ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|.

Degree: PhD, 2015, University of Cambridge

Kronecker’s first limit formula describes the constant term in the Laurent expansion of a non-holomorphic Eisenstein series at one of its poles. Asai generalised the limit formula to Eisenstein series of level one defined for a number field with class number one and obtained a function analogous to the logarithm of the absolute value of the eta function. In this thesis we reformulate Asai’s function adelically using the theory of admissible representations for GL2 and simultaneously remove the restriction on class number and level. As an application of the method, we give explicit computations of the Rankin-Selberg integral with two Eisenstein series and a cusp form.

Subjects/Keywords: Algebraic number theory; Automorphic forms; Asai's function; Eisenstein series; Kronecker Limit Formula; L-functions; Rankin-Selberg integral

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chen, C. (2015). ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/306109https://www.repository.cam.ac.uk/bitstream/1810/306109/2/license.txt

Chicago Manual of Style (16^{th} Edition):

Chen, Cangxiong. “ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|.” 2015. Doctoral Dissertation, University of Cambridge. Accessed July 15, 2020. https://www.repository.cam.ac.uk/handle/1810/306109https://www.repository.cam.ac.uk/bitstream/1810/306109/2/license.txt.

MLA Handbook (7^{th} Edition):

Chen, Cangxiong. “ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|.” 2015. Web. 15 Jul 2020.

Vancouver:

Chen C. ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|. [Internet] [Doctoral dissertation]. University of Cambridge; 2015. [cited 2020 Jul 15]. Available from: https://www.repository.cam.ac.uk/handle/1810/306109https://www.repository.cam.ac.uk/bitstream/1810/306109/2/license.txt.

Council of Science Editors:

Chen C. ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|. [Doctoral Dissertation]. University of Cambridge; 2015. Available from: https://www.repository.cam.ac.uk/handle/1810/306109https://www.repository.cam.ac.uk/bitstream/1810/306109/2/license.txt

2.
Cho, Jaehyun.
Automorphic *L*-functions and their applications to Number Theory.

Degree: 2012, University of Toronto

URL: http://hdl.handle.net/1807/32684

The main part of the thesis is applications of the Strong Artin conjecture to number theory. We have two applications. One is generating number fields with extreme class numbers. The other is generating extreme positive and negative values of Euler-Kronecker constants.
For a given number field K of degree n, let K̂ be the normal closure of K with Gal(K̂/\Bbb Q)=G. Let Gal(K̂/K)=H for some subgroup H of G. Then,
L(s,ρ,K̂/\Bbb Q)=\frac{ζ_{K}(s)}{ζ(s)}
where Ind_{H}^{G1}_{H} = 1_{G} + ρ.
When L(s,ρ) is an entire function and has a zero-free region [α,1] × [-(log N)^{2}, (log N)^{2}] where N is the conductor of L(s,ρ), we can estimate log L(1,ρ) and \frac{L'}{L}(1,ρ) as a sum over small primes:
log L(1,ρ) = ∑_{p ≤ (log N)k}λ(p)p^{-1} + O_{l,k,α}(1)
\frac{L'}{L}(1,ρ)=-∑_{p ≤ x} \frac{λ(p) log{p}}{p} +O_{l,x,α}(1).
where 0 < k < \frac{16}{1-α} and (log N)^{\frac{16}{1-α}} ≤ x ≤ N^{\frac{1}{4}}. With these approximations, we can study extreme values of class numbers and Euler-Kronecker constants.
Let \frak{K} (n,G,r_{1},r_{2}) be the set of number fields of degree n with signature (r_{1},r_{2}) whose normal closures are Galois G extension over \Bbb Q. Let f(x,t) ∈ \Bbb Z[t][x] be a parametric polynomial whose splitting field over \Bbb Q (t) is a regular G extension. By Cohen's theorem, most specialization t∈ \Bbb Z corresponds to a number field K_{t} in \frak{K} (n,G,r_{1},r_{2}) with signature (r_{1},r_{2}) and hence we have a family of Artin L-functions L(s,ρ,t). By counting zeros of L-functions over this family, we can obtain L-functions with the zero-free region above.
In Chapter 1, we collect the known cases for the Strong Artin conjecture and prove it for the cases of G=A_{4} and S_{4}. We explain how to obtain the approximations of log (1,ρ) and \frac{L'}{L}(1,ρ) as a sum over small primes in detail. We review the theorem of Kowalski-Michel on counting zeros of automorphic L-functions in a family.
In Chapter 2, we exhibit many parametric polynomials giving rise to regular extensions. They contain the cases when G=C_{n}, 3 ≤ n ≤ 6, D_{n}, 3 ≤ n ≤ 5, A_{4}, A_{5}, S_{4}, S_{5} and S_{n}, n ≥ 2.
In Chapter 3, we construct number fields with extreme class numbers using the parametric polynomials in Chapter 2.
In Chapter 4, We construct number fields with extreme Euler-Kronecker constants also using the parametric polynomials in Chapter 2.
In Chapter 5, we state the refinement of Weil's theorem on rational points of algebraic curves and prove it.
The second topic in the thesis is about simple zeros of Maass L-functions. We consider a Hecke Maass form f for SL(2,\Bbb Z). In Chapter 6, we show that if the L-function L(s,f) has a non-trivial simple zero, it has infinitely many simple zeros. This result is an extension of the result of Conrey and Ghosh.

PhD

Subjects/Keywords: Automorphic L-function; Strong Artin Conjecture; class number; Euler-Kronecker constant; simple zero; Maass L-function; 0405

…Bibliography
102
viii
Chapter 1
Artin *L*-functions
1.1
The strong Artin Conjecture… …extension K/Q. Artin conjectured that the Artin *L*-function *L*(s, ρ) is a holomorphic… …function on the complex plane and it is called the Artin Conjecutre. Langlands conjectured that *L*… …x28;s, ρ) actually is automorphic, i.e. *L*(s, ρ) = *L*(s, π) for a… …cuspidal automorphic representation π of GL(n)/Q. Since a cuspidal automorphic *L*…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Cho, J. (2012). Automorphic L-functions and their applications to Number Theory. (Doctoral Dissertation). University of Toronto. Retrieved from http://hdl.handle.net/1807/32684

Chicago Manual of Style (16^{th} Edition):

Cho, Jaehyun. “Automorphic L-functions and their applications to Number Theory.” 2012. Doctoral Dissertation, University of Toronto. Accessed July 15, 2020. http://hdl.handle.net/1807/32684.

MLA Handbook (7^{th} Edition):

Cho, Jaehyun. “Automorphic L-functions and their applications to Number Theory.” 2012. Web. 15 Jul 2020.

Vancouver:

Cho J. Automorphic L-functions and their applications to Number Theory. [Internet] [Doctoral dissertation]. University of Toronto; 2012. [cited 2020 Jul 15]. Available from: http://hdl.handle.net/1807/32684.

Council of Science Editors:

Cho J. Automorphic L-functions and their applications to Number Theory. [Doctoral Dissertation]. University of Toronto; 2012. Available from: http://hdl.handle.net/1807/32684

3.
Constable, Jonathan A.
* Kronecker*'s Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares.

Degree: 2016, University of Kentucky

URL: https://uknowledge.uky.edu/math_etds/35

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence.
In the second chapter we introduce the class number, proper class number and complete class number as well as two refinements, which facilitate the development of a connection with binary quadratic forms.
Our third chapter is devoted to deriving several class number formulas in terms of divisors of the determinant. This chapter also contains lower bounds on the class number for bilinear forms and classifies when these bounds are attained.
Lastly, we use the class number formulas to rigorously develop Kronecker's connection between binary bilinear forms and binary quadratic forms. We supply purely arithmetic proofs of five results stated but not proven in the original paper. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss.

Subjects/Keywords: complete equivalence; binary bilinear forms; binary quadratic forms; class number relations; L. Kronecker; Gauss; Algebra; Number Theory

…Bilinear Forms with
. . . . . . . . . . . .
114
114
118
124
Chapter 3 *Kronecker* Reduced… …Via Divisors of D
4.1 Using Divisors of D to count K + *L*. . . . .
4.2 Using divisors of D to… …Continuation of Figure 3.1 showing the relationships P1 = R1 + K and
P2 = S1 + *L*… …extensively. A lesser known paper by Leopold *Kronecker*
in 1883 [Kr1897] contains a novel… …develop materials to aid our understanding of *Kronecker* reduced bilinear forms. Notable
results…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Constable, J. A. (2016). Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares. (Doctoral Dissertation). University of Kentucky. Retrieved from https://uknowledge.uky.edu/math_etds/35

Chicago Manual of Style (16^{th} Edition):

Constable, Jonathan A. “Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares.” 2016. Doctoral Dissertation, University of Kentucky. Accessed July 15, 2020. https://uknowledge.uky.edu/math_etds/35.

MLA Handbook (7^{th} Edition):

Constable, Jonathan A. “Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares.” 2016. Web. 15 Jul 2020.

Vancouver:

Constable JA. Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares. [Internet] [Doctoral dissertation]. University of Kentucky; 2016. [cited 2020 Jul 15]. Available from: https://uknowledge.uky.edu/math_etds/35.

Council of Science Editors:

Constable JA. Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares. [Doctoral Dissertation]. University of Kentucky; 2016. Available from: https://uknowledge.uky.edu/math_etds/35